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Bird's array notation
Bird's array notation is a googological notation invented by Chris Bird. It is an extension of Jonathan Bowers' Extended Array Notation, and is akin to BEAF both in its history and its definition. Linear and multidimensional arrays For linear and multidimensional arrays, BAN is the same as BEAF. *Rule 1: With one or two entries, we have \(\{a\} = a\), \(\{a,b\} = a^b\). *Rule 2: If the last entry is 1, it can be removed: \(\{\# 1\} = \{\#\}\). (The octothorpe indicates the unchanged remainder of the array.) *Rule 3: If the second entry is 1, the value is just the first entry: \(\{a,1 \#\} = a\). *Rule 4: If the third entry is 1: *:\(\{a,b,1,1,\cdots,1,1,c \#\} = \{a,a,a,a,\cdots,a,\{a,b-1,1,1,\cdots,1,1,c \#\},c-1 \#\}\) *Rule 5: Otherwise: *:\(\{a,b,c \#\} = \{a,\{a,b-1,c \#\},c-1 \#\}\) With multidimensional arrays, Bird uses \(\textrm` a \langle c \rangle b \textrm'\), which is equivalent to Bowers' \(b^c \& a\). These strings are written within the quote signs and have their own specific rules: * Rule A1: If \(c = 0\), we have \(\textrm` a \langle 0 \rangle b = a \textrm'\). * Rule A2: If \(b = 1\), we have \(\textrm` a \langle c \rangle 1 = a \textrm'\). * Rule A3: Otherwise, \(\textrm` a \langle c \rangle b \textrm' = \textrm` a \langle c - 1 \rangle b c a \langle c \rangle (b - 1) \textrm'\). The main rules are: * Rule M1: If there are only two entries, \(\{a, b\} = a^b\). * Rule M2: If \(m < n\), we have \(\{\# m 1 n \#\} = \{\# n \#\}\). (This also removes ones from the end of an array.) * Rule M3: If the second entry is 1, we have \(\{a,1 \#\} = a\). * Rule M4: If there is a non-zero entry immediately after batch of unfilled separators: *:\(\{a,b m_1 1 m_2 \cdots 1 m_x c \#\} = \{a \langle m_1 \rangle b m_1 a \langle m_2 \rangle b m_2 \cdots a \langle m_x \rangle b m_x (c-1) \#\}\) * Rule M5: If there is a non-zero entry after batch of unfilled separators and string of 1's. *:\(\{a,b m_1 1 m_2 \cdots 1 m_x 1,1,\cdots,1,1,c \#\} = \{a \langle m_1 \rangle b m_1 a \langle m_2 \rangle b m_2 \cdots a \langle m_x \rangle b m_x a,a,\cdots,1,1,c-1 \#\}\) * Rule M6: If there is a string of ones starting on the first entry: *:\(\{a,b,1,1,\cdots,1,1,c \#\} = \{a,a,a,a,\cdots,a,\{a,b-1,1,1,\cdots,1,1,c-1 \#\},c-1 \#\}\) * Rule M7: Rules M1-M6 don't apply. *:\(\{a,b,c \#\} = \{a,\{a,b-1,c \#\},c-1 \#\}\) Bird uses \(m\) as a dimensional separator; in Bowers' notation it is equivalent to an \((m - 1)\) separator. This resolves a minor issue in BEAF, where ones are default in the array and zeroes are default in the separators. Hyperdimensional and nested arrays After that, Bird also goes to transform the number inside [] to the array. He denotes 1,2 to indicate the 1st separator in the 2nd 2-superdimension, m,2 to indicate the mth separator in the 2nd 2-superdimension, m,n to indicate the mth separator in the nth 2-superdimension. Next follows separators like 1,1,2, 1,1,1,2, and so on. Rules M1-M7 remains unchanged, except that for every mn we need to append #: mn is replaced to #. Only one new rule is added to the angle bracket rules. The old rule A3 becomes A4 and applies when the rules A1-A3 don't apply, and a new rule A3 is created as follows: * Rule A3. Condition: 1st entry in the angle brackets is 0, and there exists a non-zero entry after it. : \( 'a < 0,1,1,\cdots,1,1,c \# > b' = 'a < b,b,b,\cdots,b,b,c-1 \# > b' \) That rule is very similar to the Rule M4 except that 0 and 1's are filled by b's instead of a's, and the last 1 isn't replaced by the almost identical copy of the initial array. Next, we can have strings inside arrays and thereby define nested array notation. Now the rule A3 becomes A4, A4 becomes A5 and the new rule A3 is added and A4 is generalized: * Rule A3. Condition: A<B. : \('a <\# A 1 B \#> b' = 'a <\# B \#> b'\) * Rule A4. Condition: 1st entry in the angle brackets is 0, and there exists non-zero entry after it. : \('a <0 A_1 1 A_2 \cdots 1 A_n c \#> b' = 'a b A_1 b b A_2 \cdots b b A_n c-1 \#> b'\) An and B are arrays and Ai' is identical to Ai except the first entry is reduced by 1. External links *Chris Bird's Super Huge Numbers Category:Notations